2.268   ODE No. 268

\[ f(x) y(x) y'(x)+g(x) y(x)^2+h(x)=0 \] Mathematica : cpu = 0.171931 (sec), leaf count = 146


\[\left \{\left \{y(x)\to -\exp \left (\int _1^x-\frac {g(K[1])}{f(K[1])}dK[1]\right ) \sqrt {2 \int _1^x-\frac {\exp \left (-2 \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\right ) h(K[2])}{f(K[2])}dK[2]+c_1}\right \},\left \{y(x)\to \exp \left (\int _1^x-\frac {g(K[1])}{f(K[1])}dK[1]\right ) \sqrt {2 \int _1^x-\frac {\exp \left (-2 \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\right ) h(K[2])}{f(K[2])}dK[2]+c_1}\right \}\right \}\] Maple : cpu = 0.074 (sec), leaf count = 118


\[y \relax (x ) = {\mathrm e}^{\int -\frac {2 g \relax (x )}{f \relax (x )}d x} \sqrt {{\mathrm e}^{2 \left (\int \frac {g \relax (x )}{f \relax (x )}d x \right )} \left (-2 \left (\int \frac {{\mathrm e}^{\int \frac {2 g \relax (x )}{f \relax (x )}d x} h \relax (x )}{f \relax (x )}d x \right )+c_{1}\right )}\]